#8613275

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26,31,35)(27,32,38,37)(28,39,29,33)(30,34,40,36);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(25,27)(26,32)(28,34)(29,36)(30,33)(31,38)(35,37)(39,40);(2,3)(6,7)(10,16)(13,19)(17,21)(18,24)(25,28,31,29)(26,33,35,39)(27,36,38,34)(30,32,40,37);(4,8)(6,7)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,29,31,28)(26,33,35,39)(27,34,38,36)(30,32,40,37);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,26)(27,37)(28,39)(29,33)(30,34)(31,35)(32,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,30)(26,34)(27,33)(28,32)(29,37)(31,40)(35,36)(38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8520471);